Bookinglayer has an integration with Airbnb through a channel manager MyAllocator.
In order to get the corresponding rooms to sync, they have to be mapped together in MyAllocator. Rooms can be named differently in Bookinglayer and AirBnB but they need to be set under the same room types in MyAllocator.
With the help of the channel manager both Airbnb and Bookinglayer calendars should automatically be updated.
However, note that cancelled bookings in Airbnb don't update to MyAllocator and needs to be updated manually to Bookinglayer's calendar. If there's a problem with integration between AirBnB and MyAllocator, please turn to their side.
MyAllocator allows to set up different prices for Airbnb and Bookinglayer. This is helpful if you want to offer the accommodation with the cheaper price through your own website and with the higher price through the partner channels.